Framework and statement of the theorem

should satisfy the following conditions:

(F1)⁴ The process ĎX = (X^◦, X) belongs to H²(G;R)× H^2,d(G;R) and the pair (X, C) satisfiesĎ Assumption (C).

(F2) The terminal condition satisfiesξ∈L²_βˆfor some ˆβ >0.

(F3) The generator⁵ of the equation f : Ω×R+ ×Rⁿ ×R^n× ×H −→ Rⁿ is such that for any (y, z, u)∈Rⁿ×R^n××H, the map

(ω, t)−→f(ω, t, y, z, u(ω, t,·)) isFt⊗ B([0, t])−measurable.

Moreover,f satisfies a stochastic Lipschitz condition, that is to say there exist

r: (Ω×R+,P)−→(R+,B(R+)) and ϑ= (θ^c, θ^d) : (Ω×R+,P)−→(R²₊,B(R²₊)), such that, for dP⊗dC−a.e. (ω, t)∈Ω×R+

f(ω, t, y, z, u(ω, t,·))−f(ω, t, y, z, u(ω, t,·))²₂

≤rt(ω)y−y²₂+θ^c_t(ω)ct(ω)(z−z)²₂+θ_t^d(ω)

|||ut(·)−u_t(·)|||t(ω)₂

. (II.16) (F4) Let⁶α²_· := max{√r_·, θ^c_·, θ^d_·} and define the increasing,G−predictable and c`adl`ag process

A_·:=

(0,·]α²_sdCs. (II.17)

Then there exists Φ>0 such that

ΔAt(ω)≤Φ, for dP⊗dC−a.e.(ω, t)∈Ω×R+. (II.18) (F5) We have for the same ˆβ as in(F2)

E

*

(0,T]

e^{βAˆ t}f(t,0,0,0)²₂ α²_t dCt

+

<∞, where 0denotes the null application fromRto R.

Remark II.9. In the case where the integrator C of the Lebesgue–Stieltjes integral is a continuous process, we can choose between the integrands

f(t, Yt, Zt, U(t,·))

t∈0,T and

f(t, Yt−, Zt, U(t,·))

t∈0,T,

and we still obtain the same solution, as they coincide on a dP⊗dC−null set. However, in the case where the integrator C is c`adl`ag, the corresponding solutions may differ. In the formulation of the problem we have chosen the first one, while all our results can readily be adapted to the second case as well.

Recall, however, that in Section II.2 we have seen that, in special cases, the conditions for existence and uniqueness of a solution in these two cases can differ significantly.

In classical results on BSDEs, the pair (ξ, f) is called standard data. In our case, the following definition generalises this term.

Definition II.10. We will say that the sextuple (G, T,ĎX, ξ, C, f) is thestandard data underβ, wheneverˆ its elements satisfy Assumptions (F1)–(F5)for this specific ˆβ.

Definition II.11. Asolution of the BSDE (II.1) with standard data (G, T,X, ξ, C, fĎ ) under ˆβ >0 is a quadruple of processes

(Y, Z, U, N)∈H²β×H²β^,◦×H²β^,× H²β^,⊥ or (Y, Z, U, N)∈ Sβ²×H²β^,◦×H²β^,× H²β^,⊥, for some β≤βˆsuch that,P−a.s., for anyt∈0, T,

Yt=ξ+

(t,T]

f(s, Ys, Zs, U(s,·))dCs− T

t

Z_sdX_s^c− T

t

Rⁿ

U(s, x)!μ(ds,dx)− T

t

dNs. (II.1)

4We simply restate the black square above in order to have a convenient reference for it.

5This is also calleddriver of the BSDE.

6We assume, without loss of generality, thatαt>0, dP⊗dC−a.e.

II.4. FRAMEWORK AND STATEMENT OF THE THEOREM 59

Remark II.12. We emphasise that in (II.1), the stochastic integrals are well defined since (Z, U, N)∈ H²β^,◦×H²β^,× H²β^,⊥. Let us verify that the integral

(0,·]f(s, Ys, Zs, U(s,·))dCs

is also well-defined. First of all, we know by definition that for any (y, z, u)∈Rⁿ×R^n××H, there exists a dP⊗dC−null setN^y,z,u such that for any (ω, t)∈ N/ ^y,z,u

f(ω, t, y, z, u(ω, t,·)) is well defined and u(ω, t,·)∈Hω,t.

Moreover, by Lemma II.7, we know also that for some dP⊗dC−null setN!, we have for every (ω, t)∈/N!, that Hω,t is Polish for the norm|||·|||t(ω), so that it admits a countable dense subset which we denote by Hω,t. Let us then define

H :=

&

u∈H, u(ω, t,·)∈Hω,t, ∀(ω, t)∈/ N!'

, N :=.

N^y,z,u, (y, z, u)∈Q^d×Q^n××H , where QandQ^n×are the subsets of Rand R^n× with rational components.

Then, since H is countable, N is still a dP⊗dC−null set. Then, it suffices to use (F3) to re-alize that for any (ω, t) ∈ N ∪/ N!, f is continuous in (y, z, u), and conclude that we can actually de-fine f(ω, t, y, z, u(ω, t,·)) outside a universal dP⊗dC−null set. This implies in particular that for any (Y, Z, U)∈H²β×H²β^,◦×H²β^,

f(ω, t, Yt(ω), Zt(ω), U(ω, t,·)) is defined for dP⊗dC−a.e.(ω, t)∈Ω×0, T. Finally, it suffices to use(F3)and(F5)to conclude that

(0,T]

f(ω, t, Yt(ω), Zt(ω), U(ω, t,·))2dCt(ω) is finiteP−a.s..

II.4.1. Existence and uniqueness: statement. We devote this subsection to the statement of our theorem. Before that, we need some preliminary results of a purely analytical nature, whose proofs are relegated to Appendix A.1.

Lemma II.13. Fixβ,Φ>0and consider the setCβ:={(γ, δ)∈(0, β]², γ < δ}. We define the following quantity

Π^Φ(γ, δ) := 9

δ+ (2 + 9δ)e^(δ−γ)Φ γ(δ−γ). Then, the infimum of Π^Φ is given by

M^Φ(β) := inf

(γ,δ)∈Cβ

Π^Φ(γ, δ) = 9

β + Φ²(2 + 9β) /β²φ²+ 4−2exp

βΦ + 2−/

β²Φ²+ 4 2

, and is attained at the point

γ^Φ(β), β where

γ^Φ(β) := βΦ−2 +/

4 +β²Φ²

2Φ .

In addition, if we define

Π^Φ(γ, δ) := 8 γ +9

δ+ 9δe^(δ−γ)Φ γ(δ−γ),

then the infimum of Π^Φ is given by M^Φ(β) := inf₍γ,δ)∈CβΠ^Φ(γ, δ) = Π^Φ(γ^Φ(β), β), where γ^Φ(β)is the unique solution in(γ^Φ(β), β)of the equation with unknownx

8(β−x)²−9βe^(β−x)Φ

Φx²−(βΦ−2)x−β

= 0.

Moreover, it holds

βlim→∞M^Φ(β) = lim

β→∞M^Φ(β) = 9eΦ.

Theorem II.14. Let (G, T,X, ξ, C, fĎ ) be standard data under βˆ. If M^Φ( ˆβ)< ¹₂ (resp. M^Φ( ˆβ)< ¹₂), then there exists a unique quadruple (Y, Z, U, N) which satisfies (II.1) and with (Y, Z, U, N)βˆ < ∞ (resp. (Y, Z, U, N),ˆβ<∞).

Remark II.15. Using the results of Lemma II.13 and Theorem II.14, it is immediate that as soon as Φ< 1

18e, (II.19)

then there always exists a unique solution of the BSDE for ˆβ large enough.

Moreover, let us now argue why the above condition rules out the counterexamples of Section II.2 from our setting. The generator f(t, y, u) = ¹_p(y +g(u)) needs to be Lipschitz so that it fits in our framework, and to satisfy (II.16). Let us further assume that the function g is also Lipschitz, say with associated constant L^g.Then, using Young’s Inequality, we can obtain

|f(t, y, u)−f(t, y⁰, u⁰)|²≤1 +ε

p² |y−y⁰|²+ 1

p² 1 +(L^g)² ε

|u−u⁰|², for everyε >0.

Therefore, we have thatα²_r = max√

1 +ε/p, 1 +^(L_ε^g)2

/p² , and it holds α²_r∆C_r = max√

1 +ε, 1 +(L^g)² ε

/p ≥√

1 +ε > 1

18e for every ε >0.

II.5. A priori estimates

The method of proof we will use follows and extends the one of El Karoui and Huang [29]. In [29], as well as in Pardoux and Peng [55], the result is obtained using fixed-point arguments and the so-called a priori estimates. However, we would like to underline that the proof of such estimates in our case is significantly harder, due to the fact that the processC is not necessarily continuous.

The following result can be seen as the a priori estimates for a BSDE whose generator does not depend on the solution. In order to keep notation as simple as possible, as well as to make clearer the link with the data of the problem we consider, we will reuse part of the notation of(F1)–(F5), namely ξ, T, f, C, α andA, only for the next two lemmata.

Lemma II.16. Let y be an n−dimensional G−semimartingale of the form y_t=ξ+

Z

(t,T]

f_sdC_s− Z T

t

dη_s, (II.20)

whereT is aG−stopping time,ξ∈L²(G_T;Rⁿ),f is ann−dimensional optional process, C∈ V_pred⁺ (G;R) andη∈ H²(G;Rⁿ).

Let A:=α²·C for some predictable process α. Assume that there exists Φ >0 such that property (II.18) holds forA. Suppose there existsβ ∈R+ such that

E

e^βATkξk²₂ +E

"

Z

(0,T]

e^βAtkftk²₂ α²_t dC_t

#

<∞.

Then we have for any(γ, δ)∈(0, β]², withγ6=δ, kαyk²

H²_δ≤ 2e^δΦ δ kξk²

L²_δ+ 2Λ^γ,δ,Φ f α

2

H²_γ∨δ

, kyk²_S2 δ

≤8kξk²

L²_δ+8 γ f α

2

H²_γ∨δ

,

kηk²_H2

δ≤9 1 + e^δΦ kξk²

L²δ+ 9 1

γ∨δ+δΛ^γ,δ,Φ f α

2

H²γ∨δ

, where we have defined

Λ^γ,δ,Φ:= 1∨e^(δ−γ)Φ γ|δ−γ| . As a consequence, we have

kαyk²_H2 δ

+kηk²_H2

δ≤Πe^δ,Φkξk²_L2 δ

+ Π^Φ(γ, δ) f α

2

H²γ∨δ

, (II.21)

kyk²_S2 δ

+kηk²_H2 δ

≤Πe^δ,Φ_? kξk²

L²_δ+ Π^Φ_?(γ, δ) f α

2

H²γ∨δ

, (II.22)

II.5.A PRIORIESTIMATES 61

where

Πe^δ,Φ:= 9 + (9 +2

δ)e^δΦ and Πe^δ,Φ_? := 17 + 9e^δΦ. Proof. Recall the identity

y_t=ξ+ Z

(t,T]

f_sdC_s− Z T

t

dη_s=E

"

ξ+ Z

(t,T]

f_sdC_s

Gt

#

(II.23) and introduce the anticipating function

Ft:=

Z

(t,T]

fsdCs. (II.24)

Forγ∈R+, we have by the Cauchy–Schwarz inequality, kF_tk²₂≤

Z

(t,T]

e^−γAs dA_s Z

(t,T]

e^γAskf_sk²₂ α²_s dC_s≤

Z

(A_t,A_T]

e^−γALsds Z

(t,T]

e^γAskf_sk²₂ α²_s dC_s

≤ Z

(At,AT]

e^−γsds Z

(t,T]

e^γAskfsk²₂

α²_s dCs≤ 1 γe^−γAt

Z

(t,T]

e^γAskfsk²₂

α²_s dCs, (II.25) where for the third inequality we used Lemma I.34.(vii). Fort= 0, since we assumed that

E

"

Z

(0,T]

e^βAtkftk²₂ α²_t dC_t

#

<∞, we have thatE

|F0|²

<∞is true for 0< γ < β. Forδ∈R+ and by integrating (II.25) with respect to e^δAtdAt it follows

Z

(0,T]

e^δAtkFtk²₂dAt (II.25)

≤ 1 γ

Z

(0,T]

e^(δ−γ)At Z

(t,T]

e^γAskfsk²₂

α²_s dCsdAt

= 1 γ

Z

(0,T]

e^γAskfsk²₂ α²_s

Z

(0,s)

e^(δ−γ)AtdAtdCs≤ 1 γ

Z

(0,T]

e^γAskfsk²₂ α²_s

Z

(0,s]

e^(δ−γ)AtdAtdCs, (II.26) where we used Tonelli’s Theorem in the equality. We can now distinguish between two cases:

•Forδ > γ, we apply Corollary I.36 forg(x) = e^(δ−γ)x, and inequality (II.26) becomes Z

(0,T]

e^δAtkF_tk²₂dA_t≤ e^(δ−γ)Φ γ

Z

(0,T]

e^γAskfsk²₂ α²_s

Z

(A₀,A_s]

e^(δ−γ)tdtdC_s

≤ e^(δ−γ)Φ γ(δ−γ)

Z

(0,T]

e^δAskfsk²₂

α²_s dCs, (II.27)

which is integrable ifδ≤β.

•Forδ < γ, inequality (II.26) can be rewritten as follows Z

(0,T]

e^δAtkFtk²₂dAt ≤ 1 γ

Z

(0,T]

e^γAskfsk²₂ α²_s

Z

(A0,As]

e^(δ−γ)ALtdtdCs Lem. I.34.(vii)

≤ 1

γ Z

(0,T]

e^γAskfsk²₂ α²_s

Z

(A0,As]

e^(δ−γ)tdtdCs

≤ 1

γ|δ−γ|

Z

(0,T]

e^γAskfsk²₂ α²_s

e^(δ−γ)A0−e^(δ−γ)As dCs

≤ 1

γ|δ−γ|

Z

(0,T]

e^γAskfsk²₂

α²_s dCs, (II.28)

which is integrable ifγ≤β.

To sum up, forγ, δ∈(0, β], γ6=δ, we have E

"

Z

(0,T]

e^δAtkFtk²₂dAt

#

≤Λ^γ,δ,Φ f α

2

H²γ∨δ

. (II.29)

For the estimate of kαyk_H2

δ we first use the fact that kαyk²

H²δ = E

"

Z

(0,T]

e^δAtkytk²₂dAt

#

≤2E

"

Z

(0,T]

E

e^δAtkξk²₂+ e^δAtkFtk²₂ Gt

dAt

#

= 2E

"

Z

(0,∞)

E

e^δAtkξk²₂+ e^δAtkFtk²₂ Gt

dA^T_t

#

Lem. I.23

= 2E

"

Z

(0,∞)

e^δAtkξk²₂+ e^δAtkF_tk²₂dA^T_t

#

= 2E

"

Z

(0,T]

e^δAtkξk²₂+ e^δAtkF_tk²₂dA_t

#

Cor. I.36

≤

(II.29)

2e^δΦ δ kξk²

L²δ+ 2Λ^γ,δ,Φ f α

2 H²_γ∨δ

. (II.30)

In the second equality we have used that the processeskξk²₂1Ω×[0,∞](·) andkF_·k²₂are uniformly integrable, hence their optional projections are well defined. Indeed, using (II.25) and remembering thatE[kF₀k²₂]<

∞, we can conclude the uniform integrability ofkF_·k²₂. Then, by Theorem I.22 it holds that Π^G_o e^δA·kξk²₂+ e^δA·kF_·k²₂

t= e^δAtΠ^G_o kξk²₂1Ω×[0,∞](·)

t+ e^δAtΠ^G_o kF_·k²₂

t

= e^δAtE kξk²₂

Gt

+ e^δAtE kFtk²₂

Gt

=E

e^δAtkξk²₂+ e^δAtkFtk²₂ Gt

, which justifies the use of Lemma I.23.

For the estimate of kyk_S²

δ we have kyk_S2

δ =E

sup

0≤t≤T

e^δ2Atky_tk₂2

≤2E

"

sup

0≤t≤TE q

e^δAtkξk²₂+ e^δAtkF_tk²₂

G_t ²#

≤2E



 sup

0≤t≤TE

" s

e^δATkξk²₂+1

γe^(δ−γ)At Z

(t,T]

e^γAskfsk²₂ α²_s dC_t

Gt

#²



≤2E



 sup

0≤t≤TE

" s

e^δATkξk²₂+1 γ

Z

(0,T]

e^{(γ+(δ−γ)+)As}kfsk²₂ α²_s dC_s

G_t

#²



≤8E

"

e^δATkξk²₂+ 1 γ

Z

(0,T]

e^(γ∨δ)Askfsk²₂ α²_s dCs

#

≤8kξk²_L2 δ+8

γ f α

2

H²γ∨δ

(II.31) forγ∨δ≤β, where in the second and third inequalities we used the inequalitya+b≤p

2(a²+b²) and (II.25) respectively.

What remains is to controlkηk_H²

δ. We remind once more the reader thatRT

t dη_s=ξ−yt+F_t,hence E

kξ−y_t+F_tk² G_t

=E

"

Z

(t,T]

dTr[hηi_s]

G_t

#

. (II.32)

In addition, we have Z

(0,T]

e^δAsdTrhηi_s = Z

(0,T]

Z

(A0,As]

δe^δtdtdTr[hηi_s] + Tr[hηi_T]

Lem. I.34.(vii)

≤ Z

(0,T]

Z

(A₀,A_s]

δe^δALtdtdTr[hηis] + Tr[hηiT]

= δ

Z

(0,T]

Z

(0,s]

e^δAtdAtdTr[hηis] + Tr[hηiT]

≤ δ

Z

(0,T]

e^δAt Z

(t,T]

dTr[hηis]dA_t+ Tr[hηiT], (II.33) so that

kηk_H²

δ≤δE

"

Z

(0,T]

e^δAt Z

(t,T]

dTr[hηis] dA_t

# +E

Tr[hηiT]

. (II.34)

II.5.A PRIORIESTIMATES 63

For the first summand on the right-hand-side of (II.34), we have E

"

Z

(0,T]

e^δAt Z

(t,T]

dTr[hηis] dAt

#

Lemma I.23

= E

"

Z

(0,T]

e^δAtE

"

Z

(t,T]

dTr[hηis]

Gt

# dAt

#

(II.32)

= E

"

Z

(0,T]

e^δAtE

kξ−y_t+F_tk² Gt

dA_t

#

(II.32)

≤ 3E

"

Z

(0,T]

e^δAtE

kξk²₂+ky_tk²₂+kF_tk²₂ G_t

dA_t

#

(II.23)

≤ 3E

"

Z

(0,T]

e^δAtkξk²₂dAt

# + 3E

"

Z

(0,T]

e^δAtkFtk²₂dAt

#

+ 6E

"

Z

(0,T]

e^δAtE

kξk²₂+kFtk²₂ Gt

dAt

#

Lemma I.23

= 9E

"

Z

(0,T]

e^δAtkξk²₂dAt

# + 9E

"

Z

(0,T]

e^δAtkFtk²₂dAt

#

Corollary I.36

≤

(II.29)

9e^δΦ δ kξk²

L²_δ+ 9Λ^γ,δ,Φ f α

2

H²γ∨δ

.

We now need an estimate for E hR

(0,T]dTr[hηi_s]i

,i.e. the second summand of (II.34), which is given by E[Tr[hηiT]] = E

kξ−y0+F0k²

≤3E

kξk²₂+ky0k²+|F0|²

(II.23)

≤ 9E kξk²₂

+ 9E

|F0|^2(II.25)

≤ 9kξk²_L2

δ+ 9

γ∨δ f α

2

H²γ∨δ

, where we used the fact thatE

ky₀k²

≤2E

kξk²₂+|F₀|² . Then (II.34) yields

kηk²_H2

δ≤9 1 + e^δΦ kξk²

L²δ+ 9 1

γ∨δ+δΛ^γ,δ,Φ f α

2

H²γ∨δ

. (II.35)

Remark II.17. An alternative framework can be provided if we define the norms in Section II.3 using another positive and increasing function hinstead of the exponential function. In order to obtain the required a priori estimates, we need to assume that h is sub–multiplicative⁷ and that it shares some common properties with the exponential function. However, we need to assume that the process A defined in (F4)is P−a.s. bounded by a positive constant. We provide the detailed calculation for the caseh(x) = (1 +x)^ζ, forx∈R+ andζ≥1 in Appendix A.1.1.

Now we are going to state in a convenient way for later use the pathwise estimates we have obtained in Lemma II.16. These estimates will allow us to prove in Chapter IV that specific sequences are uniformly integrable. We will not need all of them in Chapter IV, however we provide all the available information we have.

Lemma II.18. Letξ, T, C, α,AandΦas in Lemma II.16. Assume that then−dimensionalG−semimartingales y¹_t andy²_t can be decomposed as follows

yⁱ_t=ξ+ Z

(t,T]

f_sⁱdC_s− Z T

t

dη_sⁱ fori= 1,2 (II.36)

wheref¹, f² aren−dimensionalG−optional processes such that E

"

Z

(0,T]

e^βAtkf_tⁱk²₂ α²_t dCt

#

<∞,

7In the proof of Sato [64, Proposition 25.4] we can find a convenient tool for constructing sub–multiplicative functions.

fori= 1,2 and for someβ ∈R+ andη¹, η²∈ H²(G;Rⁿ). Then, forγ, δ∈(0, β], with γ6=δ, Z

(0,T]

e^βAtky_tⁱk²₂dAt≤ 2

βe^βΦe^βAT sup

t∈[0,T]E h

ξⁱ

2 2

Gt

i

+ 2

γ(β−γ)e^(β−γ)Φe^(β−γ)AT sup

t∈[0,T]E Z

(0,T]

e^βAt f_sⁱ

2 2

α²_s dCs

Gt

(II.37) and

Z

(0,T]

e^βAtky¹_t−y_t²k²₂dAt≤ e^(β−γ)Φ

γ(β−γ)e^(β−γ)AT sup

t∈[0,T]

E Z

(0,T]

e^βAt

f_s¹−f_s²

2 2

α²_s dCs

Gt

(II.38) Moreover, for the martingale partsη¹, η²∈ H²(G;Rⁿ)of the aforementioned decompositions, we have

sup

t∈[0,T]

η¹_t−η¹₀

2

2≤6 sup

t∈[0,T]E

ξ¹

2 2+ 1

β Z

(0,T]

e^βAs f_s¹

2 2

α²_s dC_s

G_t

+ 3 Z

(0,T]

e^βAt f_s¹

2 2

α²_s dC_s (II.39) and

sup

t∈[0,T]

(η¹_t−η²_t)−(η₀¹−η₀²) ₂

≤ 6 β

Z

(0,T]

e^βAt

f_s¹−f_s²

2 2

α²_s dCs+3 β sup

t∈[0,T]E Z

(0,T]

e^βAs

f_s¹−f_s²

2 2

α²_s dCs

Gt

.

(II.40)

Proof. For the following assumeγ, δ∈(0, β] with γ6=δ.

• We will prove Inequality (II.37) for i= 1 by following analogous to Lemma II.16 calculations. The sole difference will be that we are going to apply the conditional form of the Cauchy–Schwartz Inequality.

Moreover, by Identity (II.36), we have y_t¹

2 2=

E

ξ¹+

Z

(t,T]

f_s¹dCs

Gt

2 2

. (II.41)

In view of these comments, we have Z

(0,T]

e^βAtky_t¹k²₂dA_t Z

(0,T]

e^βAt E

ξ¹+

Z

(t,T]

f_s¹dC_s

Gt

2 2

dA_t

≤ 2 Z

(0,T]

e^βAt E

ξ¹ Gt

2

2dA_t+ 2 Z

(0,T]

e^βAt E

Z

(t,T]

f_s¹dC_s

Gt

2 2

dA_t

C-S Ineq.

≤ 2 Z

(0,T]

e^βAt E

ξ¹ G_t

2

2dA_t+ 2 Z

(0,T]

e^βAtE h1

γe^−γAt G_ti

E Z

(t,T]

e^γAt f_s¹

2 2

α²_s dC_s

G_t

dA_t

≤ 2 Z

(0,T]

e^βAtE h

ξ¹

2 2

Gt

i

dAt+ 2 γ

Z

(0,T]

e^(β−γ)AtE Z

(t,T]

e^γAt f_s¹

2 2

α²_s dCs

Gt

dAt

≤ 2 sup

t∈[0,T]E h

ξ¹

2 2

Gt

iZ

(0,T]

e^βAtdAt+2 γ

Z

(0,T]

e^(β−γ)AtE Z

(0,T]

e^βAt f_s¹

2 2

α_s² dCs

Gt

dAt Cor. I.36

≤ 2 sup

t∈[0,T]

E h

ξ¹

2 2

G_tiZ

(0,T]

e^βAtdA_t + 2

γ sup

t∈[0,T]E Z

(0,T]

e^βAt f_s¹

2 2

α²_s dCs

Gt

Z

(0,T]

e^(β−γ)AtdAt Cor. I.36

≤ 2

βe^βΦe^βAT sup

t∈[0,T]E h

ξ¹

2 2

Gt

i

+ 2

γ(β−γ)e^(β−γ)Φe^(β−γ)AT sup

t∈[0,T]E Z

(0,T]

e^βAt f_s¹

2 2

α²_s dC_s

Gt

• We will prove Inequality (II.38). We will follow analogous arguments as in the previous case, but we are going to use instead of (II.41) the identity

ky¹_t−y_t²k²₂= E

Z

(t,T]

f_s¹−f_s²dCs

Gt

2 2

. (II.42)

II.5.A PRIORIESTIMATES 65

Now we have Z

(0,T]

e^βAtky¹_t−y_t²k²₂dAt (II.42)

= Z

(0,T]

e^βAt E

Z

(t,T]

f_s¹−f_s²dCs

Gt

2 2

dAt C-S Ineq.

≤ Z

(0,T]

e^βAtE h1

γe^−γAt Gt

i E

Z

(t,T]

e^γAt

f_s¹−f_s²

2 2

α²_s dCs

Gt

dAt

≤ 1

γ Z

(0,T]

e^(β−γ)AtE Z

(0,T]

e^βAt

f_s¹−f_s²

2 2

α²_s dCs

Gt

dAt Cor. I.36

≤ e^(β−γ)Φ

γ(β−γ)e^(β−γ)AT sup

t∈[0,T]E Z

(0,T]

e^βAt

f_s¹−f_s²

2 2

α²_s dC_s

Gt

• Now we are going to prove (II.39) fori= 1. We use initially the analogous to Inequality (II.25) in order to obtain

Z

(0,T]

f_s¹dCs

2 2

≤ 1 β

Z

(0,T]

e^βAt f_s¹

2 2

α²_s dCs. (II.43)

Moreover, by Identity (II.41) we obtain y_t¹

2 2

(II.41)

≤ E

ξ¹+

Z

(t,T]

f_s¹dCs

Gt

2 2

≤2E

ξ¹

2 2+

Z

(t,T]

f_s¹dCs

2 2

Gt

(II.43)

≤ 2E

ξ¹

2 2+ 1

β Z

(0,T]

e^βAs f_s¹

2 2

α²_s dC_s

Gt

and consequently

sup

t∈[0,T]

y¹_t

2

2≤2 sup

t∈[0,T]E

ξ¹

2 2+ 1

β Z

(0,T]

e^βAs f_s¹

2 2

α²_s dC_s

Gt

. (II.44)

Now, by Identity (II.36) we have that sup

t∈[0,T]

η¹_t−η¹₀

2

2= sup

t∈[0,T]

y_t¹−y₀¹+ Z

(0,T]

f_s¹dC_s

2 2

≤6 sup

t∈[0,T]

y_t¹

2 2+ 3

Z

(0,T]

f_s¹dC_s

2 2 (II.43)

≤

(II.44)

6 sup

t∈[0,T]E

ξ¹

2 2+1

β Z

(0,T]

e^βAs f_s¹

2 2

α²_s dC_s

G_t

+ 3 Z

(0,T]

e^βAt f_s¹

2 2

α²_s dC_s.

• We are going to prove, now, the Inequality (II.40). We use initially the analogous to Inequality (II.25) in order to obtain

Z

(0,T]

(f_s¹−f_s²) dCs

2 2

≤ 1 β

Z

(0,T]

e^βAt

f_s¹−f_s²

2 2

α²_s dCs. (II.45)

Moreover, by Identity (II.42) we have by Conditional Cauchy-Schwartz Inequality (analogously to the second case)

sup

t∈[0,T]

y¹_t−y²_t

2 2≤ 1

β sup

t∈[0,T]E Z

(0,T]

e^βAs

f_s¹−f_s²

2 2

α²_s dC_s

G_t

. (II.46)

By Identity (II.36), we have

(η¹_t −η²_t)−(η₀¹−η₀²) = (y_t¹−y_t²)−(y¹₀−y²₀) + Z

(0,t]

(f_s¹−f_s²) dCs. Finally, we have

sup

t∈[0,T]

(η_t¹−η_t²)−(η¹₀−η²₀)

₂≤ sup

t∈[0,T]

(y_t¹−y²_t)−(y₀¹−y₀²) + Z

(0,t]

(f_s¹−f_s²) dCs

2 2

≤ 6 sup

t∈[0,T]

y¹_t−y²_t

2 2+ 3

β Z

(0,T]

e^βAs

f_s¹−f_s²

2 2

α²_s dCs

(II.45)

≤

(II.46)

6 β

Z

(0,T]

e^βAt

f_s¹−f_s²

2 2

α²_s dC_s+ 3 β sup

t∈[0,T]E Z

(0,T]

e^βAs

f_s¹−f_s²

2 2

α²_s dC_s

Gt

.

Remark II.19. Viewing (II.20) as a BSDE whose generator does not depend on y and η, then this BSDE has a solution, which can be uniquely determined by the pair (y, η). Indeed, consider the data (G, T, ξ, f, C) and the processesαandA, which all satisfy the respective assumptions of Lemma II.16 for some ˆβ >0. Then the semimartingale

yt=E

ξ+ Z

(t,T]

fsdCs

Gt

=E

ξ+ Z

(0,T]

fsdCs

Gt

− Z

(0,t]

fsdCs, t∈R+

satisfiesy_T =ξand forη_·:=E ξ+R

(0,T]f_sdC_s G_·

yt−yT =E

ξ+ Z

(t,T]

fsdCs

Gt

−ξ=E

ξ+ Z

(0,T]

fsdCs

Gt

− Z

(0,t]

fsdCs−ξ

=E

ξ+ Z

(0,T]

fsdCs

Gt

+

Z

(t,T]

fsdCs− Z

(0,T]

fsdCs−ξ=ηt+ Z

(t,T]

fsdCs−ηT. Now, one candidate for a processX that we can choose such that (G, T, X, ξ, C, f) become standard data for any arbitrarily chosen integrator C, is the natural pair of the zero martingale (under G), which we will denote by 0. Hence, given the standard data (G, T,0, ξ, C, f) the quadruple (y, Z, U, η) satisfies the BSDE

yt=ξ+ Z

(t,T]

fsdCs− Z T

t

dηsfort∈J0, TK,

for any pair (Z, U). Assume now that there exists a quadruple (y,e Z,e U ,e η) which satisfiese ye_t=ξ+

Z

(t,T]

f_sdC_s− Z T

t

dηe_sfort∈J0, TK. Then, the pair (y−y, ηe −eη) satisfies

y−yet=− Z T

t

d η−ηe

sfort∈J0, TK

and by Lemma II.16, for ξ = 0 and f = 0, we conclude that ky−eyk_S² =kη−ηke _H² = 0. Therefore y and y, resp.e η and η, are indistinguishable, which implies our initial statement that every solution cane be uniquely determined by the pair (y, η).

In order to obtain the a priori estimates for the BSDE (II.1), we will have to consider solutions (Yⁱ, Zⁱ, Uⁱ, Nⁱ), i = 1,2, associated with the data (G, T, X, ξⁱ, C, fⁱ), i = 1,2 under ˆβ, where we also assume that f¹, f² have common r, ϑ bounds. Denote the difference between the two solutions by (δY, δZ, δU, δN),as well asδξ:=ξ¹−ξ² and

δ₂f_t:= (f¹−f²)(t, Y_t², Z_t², U²(t,·)), ψt:=f¹(t, Y_t¹, Z_t¹, U¹(t,·))−f²(t, Y_t², Z_t², U²(t,·)).

We have the identity δYt=δξ+

Z

(t,T]

ψsdCs− Z T

t

δZsdX_s^◦− Z T

t

Z

Rⁿ

δU(s,·)µ(ds,e dx)− Z T

t

dδNs. (II.47) For the wellposedness of this last BSDE we need the following lemma.

Lemma II.20. The processes Z ·

0

δZsdX_s^◦ and Z ·

0

Z

Rⁿ

δU(s,·)eµ(dt,dx) are square-integrable martingales with finite associated k · kβˆ−norms.

Proof. The square-integrability is obvious. The inequalities E[Tr[hδZ·X^◦i]]≤2E

Tr[hZ¹·X^◦i]

+ 2E

Tr[hZ²·X^◦i]

, E[Tr[hδU ?µi]]e ≤2E

Tr[hU¹?µi]e + 2E

Tr[hU²?µi]e ,

II.5.A PRIORIESTIMATES 67

together with Lemma II.5 guarantee that E

"

Z

(0,T]

e^{βAˆ t}kctδZtk²₂dCt

# +E

"

Z

(0,T]

e^{βAˆ t}|||δUt(·)|||²_t dCt

#

<∞.

Therefore, by defining Ht:=

Z t 0

δZsdX_s^◦+ Z t

0

Z

Rⁿ

δU(t,·)µe(dt,dx) + Z t

0

dδNs, (II.48)

we can treat the BSDE (II.47) exactly as the BSDE (II.20), where the martingaleH will play the role of the martingaleη.

Proposition II.21 (A priori estimates for the BSDE (II.1)). Let (G, T, X, ξⁱ, C, fⁱ), be standard data under βˆfori= 1,2. Thenψ/α∈H²βˆ and, ifM^Φ( ˆβ)<1/2, the following estimates hold

k(αδY, δZ, δU, δN)k²_βˆ≤Σe^Φ( ˆβ)kδξk²

L²βˆ

+ Σ^Φ( ˆβ)

δ2f α

2

H²_βˆ

,

k(δY, δZ, δU, δN)k²_?,βˆ≤Σe^Φ_?( ˆβ)kδξk²_L2 βˆ

+ Σ^Φ_?( ˆβ)

δ₂f α

2

H²βˆ

, where

Σe^Φ( ˆβ) := Πe^β,Φˆ

1−2M^Φ( ˆβ), Σe^Φ_?( ˆβ) := minn

Πe^β,Φ_?^ˆ + 2M_?^Φ( ˆβ),Σe^Φ( ˆβ), 8 +16

βˆΣe^Φ( ˆβ)o ,

Σ^Φ( ˆβ) := 2M^Φ( ˆβ)

1−2M^Φ( ˆβ), Σ^Φ_?( ˆβ) := minn

2M_?^Φ( ˆβ)(1 + Σ^Φ( ˆβ)), 16

βˆ(1 + Σ^Φ( ˆβ))o . Proof. For the integrability ofψ, using the Lipschitz property(F3)off¹, f², we get

kψtk²₂≤2rtkδYtk²₂+ 2θ_t^ckctδZtk²₂+ 2θ_t^d|||δUt(·)|||²_t+ 2kδ2ftk²₂. Hence by the definition of α, which implies that

r

α² ≤α² and θ^c α², θ^d

α² ≤1, (II.49)

we get kψtk²₂

α²_t ≤2

α²_tkδYtk²₂+kctδZ_tk²₂+|||δUt(·)|||²_t+|δ2f|² α²

(II.50)

≤2α²_tkδYtk²₂+ 2kctδZtk²₂+ 2|||δUt(·)|||²_t + 4

α²

kf¹(s,0,0,0)k²₂+rtkY_t²k²₂+θ_t^ckctZ_t²k²₂+θ^d_t|||δUt(·)|||²_t + 4

α²

kf²(s,0,0,0)k²₂+rtkδYtk²₂+θ^c_tkctδZ_t²k²₂+θ_t^d|||δUt(·)|||²_t

≤6 α²_tkδYtk²₂+kctδZtk²₂+|||δUt(·)|||²_t + 4

α² kf¹(s,0,0,0)k²₂+kf²(s,0,0,0)k²₂ , where, having used once more that (II.49) it follows that ^ψ_α ∈H²βˆ. Next, for thek · kβˆ−norm, we have

k(δY, δZ, δU, δN)k²βˆ=kαδYk²

H²_βˆ+kδZk²

H^2,◦_βˆ +kδUk²

H^2,\βˆ

+kδNk²

H^2,⊥_ˆ

β

(II.48)

= kαδYk²

H²βˆ

+kHk²_H2 βˆ

(II.21)

≤ Πe^β,Φˆ kδξk²

L²βˆ

+M^Φ( ˆβ) ψ α

2 H²_βˆ

≤Πe^β,Φˆ kδξk²_L2 βˆ

+ 2M^Φ( ˆβ)

kαδYk²_H2 βˆ

+kδZk²

H^2,◦βˆ

+kδUk²

H^2,\_βˆ

+ 2M^Φ( ˆβ)

δ₂f α

2

H²βˆ

≤Πe^β,Φˆ kδξk²

L²_βˆ+ 2M^Φ( ˆβ)

kαδYk²

H²_βˆ+kHk²_H2 βˆ

+ 2M^Φ( ˆβ)

δ2f α

2

H²_βˆ

.

Therefore, this implies

k(αδY, δZ, δU, δN)k²_βˆ≤Σe^Φ( ˆβ)kδξk²_L2 βˆ

+ Σ^Φ( ˆβ)

δ₂f α

2

H²βˆ

. (II.51)

We can obtaina priori estimates for the k · k_?,βˆ−norm by arguing in two different ways:

•The identity (II.47) gives k(δY, δZ, δU, δN)k²_?,βˆ=kδYk²_S2

βˆ

+kδZk²

H^2,◦_βˆ +kδUk²

H^2,\βˆ

+kδNk²

H^2,⊥_ˆ

β

(II.48)

= kδYk²_S2 βˆ

+kHk²_H2 βˆ

(II.22)

≤ Πe^β,Φ_?^ˆ kδξk²_L2 βˆ

+M_?^Φ( ˆβ) ψ α

2 H²βˆ

(II.50)

≤ Πe^β,Φ_?^ˆ kδξk²

L²_βˆ+ 2M_?^Φ( ˆβ)kαδYk²

H²_βˆ+ 2M_?^Φ( ˆβ)kHk²_H2 βˆ

+ 2M_?^Φ( ˆβ)

δ2f α

2

H²_βˆ (II.51)

≤ Πe^β,Φ_?^ˆ kδξk²

L²βˆ

+ 2M_?^Φ( ˆβ)

δ2f α

2

H²_βˆ

+ 2M_?^Φ( ˆβ)



 eΣ^Φ( ˆβ)kδξk²

L²βˆ

+ Σ^Φ( ˆβ)

δ2f α

2

H²_βˆ





=

Πe^β,Φ_?^ˆ + 2M_?^Φ( ˆβ)eΣ^Φ( ˆβ) kδξk²_L2

βˆ

+ 2M_?^Φ( ˆβ)

1 + Σ^Φ( ˆβ)

δ2f α

2

H²βˆ

.

•The identity (II.23) gives kδYk²_S2

βˆ

=E

sup

0≤t≤T

e

βˆ

2A_tkδY_tk₂²_(II.23)

≤ E

"

sup

0≤t≤TE

e

βˆ

2A_tkδξk₂+ e

βˆ 2A_t

Z

(t,T]

ψ_sdC_{s 2}

G_t ²#

(II.25)

≤ 2E

"

sup

0≤t≤TE s

e^{βAˆ t}kδξk²₂+1 βˆ

Z

(t,T]

e^{βAˆ s}kψsk²₂ α²_s dC_s

G_t ²#

≤2E

"

sup

0≤t≤TE s

e^{βAˆ T}kδξk²₂+ 1 βˆ

Z

(0,T]

e^{βAˆ s}kψsk²₂ α²_s dC_s

G_t ²#

≤8E

"

e^{βAˆ T}kδξk²₂+ 1 βˆ

Z

(0,T]

e^{βAˆ s}kψsk²₂ α²_s dCs

#

(II.50)

≤ 8kδξk²

L²_βˆ+16 βˆ





δ₂f α

2

H²_βˆ

+kαδYk²

H²_βˆ+kδZk²

H^2,◦βˆ

+kδUk²

H^2,\_βˆ



, (II.52)

where, in the second and fifth inequality we used the inequalitya+b≤p

2(a²+b²) and Doob’s inequality respectively. Then we can derive the required estimate

k(δY, δZ, δU, δN)k²_?,βˆ = kδYk²_S2 βˆ

+kδZk²

H^2,◦βˆ

+kδUk²

H^2,\_βˆ +kδNk²

H^2,⊥_ˆ

β

(II.48)

= kδYk²_S2 βˆ

+kHk²_H2 βˆ

(II.48)

≤

(II.52)

8kδξk²

L²βˆ

+16 βˆ

δ2f α

2

H²_βˆ

+16

βˆkαδYk²

H²βˆ

+16 βˆkHk²_H2

βˆ

(II.51)

≤

8 +16 βˆΣe^Φ( ˆβ)

kδξk²_L2

βˆ

+16 βˆ

1 + Σ^Φ( ˆβ)

δ₂f α

2

H²βˆ

.

II.6. Proof of the theorem

We will use now the previous estimates to obtain the existence of a unique solution using a fixed point argument. The reader should keep in mind that the fixed point argument boils the existence and uniqueness of the solution of the BSDE (II.1) down to a martingale representation problem.

II.6. PROOF OF THE THEOREM 69

Proof of Theorem II.14. Let (y, z, u, w) be such that (αy, z, u, w)∈H²βˆ×H^2,◦βˆ ×H^2,\βˆ × H^2,⊥. Then the processM defined by

M_·:=E

"

ξ+ Z

(0,T]

f(s, y_s, z_s, u(s,·)) dCs

G·

#

+w_·∈ H²(G;Rⁿ),

and by Proposition I.165 it has a unique, up to indistinguishability, orthogonal decomposition M·=M0+

Z · 0

ZsdX_s^◦+ Z ·

0

Z

R^`

U(s, x)µ(ds,e dx) +L·, where (Z, U, L)∈H^2,◦×H^2,\× H^2,⊥. In view of the identity

M_T −M_t= Z T

t

Z_sdX_s^◦+ Z T

t

Z

R^`

U(s, x)µ(ds,e dx) + Z T

t

dL_s, 0≤t≤T, we obtain

E

"

ξ+ Z

(t,T]

f(s, ys, zs, u(s,·)) dCs

Gt

#

=ξ+ Z

(t,T]

f(s, y_s, z_s, u(s,·)) dC_s− Z T

t

Z_sdX_s^◦− Z T

t

Z

R^`

U(s, x)µ(ds,e dx)− Z T

t

dN_s, where N:=L−w. Define now

Yt:=E

"

ξ+ Z

(t,T]

f(s, ys, zs, u(s,·)) dCs

Gt

# .

In order to construct a contraction using Lemma II.16, we need to chooseδ > γ. Then by Lemma II.13 we can choose γ^? ∈(0,β] such that infˆ _{(γ,δ)∈Cˆ}

βΠ^Φ(γ, δ) = Π^Φ(γ^?( ˆβ),βˆ). Now we get that (αY, Z·X^◦+ U ?µe+N)∈H²βˆ× H²_ˆ

β, and due to the orthogonality of the martingales, recall Lemma I.164, we conclude that (αY, Z, U, N)∈H²βˆ×H^2,◦βˆ ×H^2,\βˆ × H^2,⊥_ˆ

β . Hence, the operator S : H²βˆ×H^2,◦βˆ ×H^2,\βˆ × H^2,⊥_ˆ

β ,k · kβˆ

−→ H²βˆ×H^2,◦βˆ ×H^2,\βˆ × H^2,⊥_ˆ

β ,k · kβˆ

that maps the processes (αy, z, u, w) to the processes (αY, Z, U, N) defined above, is indeed well-defined.

Let (αyⁱ, zⁱ, uⁱ, wⁱ)∈H²βˆ×H^2,◦βˆ ×H^2,\βˆ × H^2,⊥_ˆ

β fori= 1,2, with S αyⁱ, zⁱ, uⁱ, wⁱ

= (αYⁱ, Zⁱ, Uⁱ, Nⁱ), for i= 1,2.

Denote, as usual,δy, δz, δu, δw the difference of the processes and ψt:=f t, y¹_t, z¹_t, u¹(t,·)

−f t, y_t², z_t², u²_t(·) . It is immediate that ^ψ_α ∈H²βˆ and that

kS αy¹, z¹, u¹, w¹

−S αy², z², u², w²

k²_βˆ=kαδYk²_H2 βˆ

+kδZk²

H^2,◦βˆ

+kδUk²

H^2,\_βˆ +kδNk²_H2,⊥

βˆ

δξ=0

≤

Lemma II.16

M^Φ( ˆβ)

ψ α

2

H²βˆ

(II.50)

≤ 2M^Φ( ˆβ)

kαδyk²_H2 βˆ

+kδzk²

H^2,◦βˆ

+kδuk²

H^2,\_βˆ

Lemma II.16

≤ 2M^Φ( ˆβ)

αy¹, z¹, u¹, w¹

− αy², z², u², w²

2 βˆ.

Hence, for M^Φ( ˆβ) <1/2, we can apply Banach’s fixed point theorem to obtain the existence of a unique fixed point (Y , Z, U, Ne ). To obtain a solution in the desirable spaces, we substitute Ye in the triplet with Y, the corresponding c`adl`ag version; indeed, G satisfies the usual conditions and Ye is a semimartingale.

The exact same reasoning using thek·k_S2 βˆ

−norm forY leads to a contraction whenM_?^Φ( ˆβ)<1/2.

Corollary II.22 (Picard approximation). Assume thatM^Φ( ˆβ)<1/2 (resp. M_?^Φ( ˆβ)<1/2) and define a sequence (Υ^(p))_p∈N on H²βˆ×H^2,◦βˆ ×H^2,\βˆ × H^2,⊥_ˆ

β (resp. onS²_ˆ

β×H^2,◦βˆ ×H^2,\βˆ × H^2,⊥_ˆ

β )such that Υ⁽⁰⁾ is the zero element of the product space andΥ^(p+1) is the solution of

Y_t^(p+1)= ξ+ Z T

t

f(s, Y_s^(p), Z_s^(p), U^(p)(s,·)) dCs

− Z T

t

Z_s^(p+1)dX_s^◦− Z T

t

Z

R^`

U^(p+1)(s, x)µ(ds,e dx)− Z T

t

dN_s^(p+1) Then

(i) The sequence (Υ^(p))_p∈N converges ink · kβˆ (resp. ink · k_?,βˆ), to the solution of the BSDE (II.1).

(ii) The following convergence holds Z^(p), U_s^(p), N^(p)

−−−−−→

p→∞ (Z, U, N), in Hβ^ˆ(X^◦)×Hβ^ˆ(X^d)×Hβ^ˆ(X^⊥).

(iii) There exists a subsequence (Υ^(pj))j∈Nwhich converges dP⊗e^βAˆ dC−a.e.

Proof. As in the proof of Theorem II.14, we obtain, for p≥1, kΥ^(p+1)−Υ^(p)k²_βˆ≤

2M^Φ( ˆβ)^p

kΥ⁽¹⁾k²_βˆ

resp. kΥ^(p+1)−Υ^(p)k²

?,βˆ≤

2M_?^Φ( ˆβ)p

kΥ⁽¹⁾k²

?,βˆ

,

(II.53) and consequently, sinceP

p∈NkΥ^(p+1)−Υ^(p)k²_ˆ

β <∞(resp. P

p∈NkΥ^(p+1)−Υ^(p)k²

?,βˆ<∞), the sequence (Υ^(p))p∈N is Cauchy under k · kβˆ (resp. under k · k_?,βˆ). Denote by Υ the unique limit on the product space. Then, it coincides with the unique fixed point for the contraction S (see the proof of Theorem II.14 above) due to the construction of (Υ^(p))p∈N, which proves (i).

For (ii), the result is immediate by the Cauchy property of the sequence (Υ^(p))p∈N, Itˆo’s isometry, the stability⁸ of the respective closed linear space generated by X^◦ and X^\ in conjunction with their orthogonality (recall Lemma I.164), which makesH^2,⊥ to be a closed subspace; see [35, Theorem 6.16].

Finally, for (iii), by the k · kβˆ−convergence, we can extract a subsequence (pj)_j∈Nsuch that

kΥ^(pj+1)−Υ^(pj)kβˆ≤2^−2j, for everyj∈N. (II.54) Define, for anyε≥0,N^p,ε:=

(ω, t)∈Ω×J0, TK, |Y_t^(p)(ω)−Yt(ω)|> ε . Then we have dP⊗e^βAˆ dC

lim sup

j→∞

N^pj,ε

= lim

j→∞dP⊗e^βAˆ dC





∞

[

i=j

h|Y^(pi)−Y|> εi





≤ lim

j→∞

1 ε²

∞

X

i=j

E

"

Z T 0

e^{βAˆ t}|Y_t^(pi)−Y_t|²dC_t

#

≤ lim

j→∞

1 ε²

∞

X

i=j

kY^(pi)−Yk²_ˆ

β

≤ lim

j→∞

1 ε²

∞

X

i=j

∞

X

m=1

2^m

Y^(pi+m+1)−Y^(pi+m)

2 H²_βˆ

!

≤ lim

j→∞

1 ε²

∞

X

m=i

∞

X

m=1

2^m

Υ^(pi+m+1)−Υ^(pi+m)

2 βˆ

!

(II.54)

≤ lim

j→∞

1 ε²

∞

X

i=j

∞

X

m=1

2^m2^−2(i+m)

!

= 0, for any ε >0.

Hence

dP⊗e^βAˆ dC

lim sup

j→∞

N^pj,0

≤X

n∈N

dP⊗e^βAˆ dC

lim sup

j→∞

N^pj,1/n

= 0.

Following the same arguments, we have the almost sure convergence ofZ^pj, U^pj, N^pj to the corre-sponding processes of the k · kβ−solution of the BSDE (II.1). Moreover, using the same steps, we can

obtain the analogous result for the k · k_?,βˆ−norm.

8At this point the stability is understood as “closed under stopping the processes”. For the precise statement see He et al. [35, Definition 6.16].

II.7. RELATED LITERATURE 71

II.7. Related literature

Let us now compare our work with the papers by Bandini [6] and Cohen and Elliott [19] who also consider BSDEs in stochastically discontinuous filtrations. The setting in [19] is rather different from ours. Indeed, in our case a driving martingale X is given right from the start, and as a consequence the process C with respect to which the generatorf is integrated is linked to the predictable bracket of X.

However, the authors of [19] do not choose anyX from the start, but consider instead a general martingale representation theorem involving countably many orthogonal martingales, which only requires the space of square integrable random variables on (Ω,F,P) to be separable to hold. Furthermore, their processC can, unlike our case, be chosen arbitrary (in the sense that it does not have to be related to the driving martingales), but with the restriction that it has to be deterministic. Moreover, it has to assign positive measure on every interval, see Definition 5.1 therein, hence C cannot be piecewise constant; the latter would naturally arise from a discrete-time martingale with independent increments, which is exactly the situation one encounters when devising numerical schemes for BSDEs. Therefore, their setting cannot be embedded into our framework, and vice versa.

On the other hand, in [6], the author considers a BSDE driven by a pure–jump martingale without an orthogonal component, which is a special case of (II.1). The martingale in this setting should actually have jumps of finite activity, hence many of the interesting models for applications in mathematical finance, such as the generalized hyperbolic, CGMY, and Meixner processes, are excluded. Such a restriction is not present at all in our framework. Otherwise, the assumptions and the conclusion in [6] are analogous to the present work. A direct comparison is however not possible, i.e. we cannot deduce the existence and uniqueness results in her work from our setting, since the assumptions are not exactly comparable.

In particular, the integrability condition (iii) on page 3 in [6] is not compatible with(F5).

Let us also compare our result with the literature on BSDEs with random terminal time. Royer [62], for instance, considers a BSDE driven by Brownian motion, where the terminal time is a [0,∞)−valued stopping time. Hence, her setting can be embedded in ours, by assuming the absence of jumps and of the orthogonal component, and further requiring that C is a continuous process. She shows existence and uniqueness of a solution under the assumptions that the generator is uniformly Lipschitz in z and continuous iny, and the terminal condition is bounded. Moreover, she requires that either the generator is strictly monotone in y and f(t,0,0) is bounded (for all t) or that the generator is monotone in y and f(t,0,0) = 0 (for allt). These conditions are not directly covered by our Assumptions(F1)–(F5), however if we consider her conditions and assume in addition that the generator is Lipschitz iny, then we can recover the existence and uniqueness result from our main theorem. Let us point out that BSDEs with constant terminal time are related to semi–linear parabolic PDEs, while BSDEs with random terminal time are associated to semi–linear elliptic PDEs.

We would also like to comment briefly on the choice of the norms we consider here. They are mostly inspired from the ones defined in the seminal work of El Karoui and Huang [29], and are equivalent to the usual norms found in the literature when the process A and timeT are both bounded. Bandini [6]

uses different spaces, where the norm is defined using the Dol´eans–Dade stochastic exponential instead of the natural exponential. In our setting whereAis allowed to be unbounded, we can only say that our norm dominates hers. This means that we require stronger integrability conditions, but as a result we will also obtain a solution of the BSDE with stronger integrability properties. In any case, our method could be adapted to this choice of the norm, albeit with modified computations in our estimates. We remind the reader Remark II.17 for a short discussion about the definition of the norms.

Let us conclude this section by commenting on the condition (II.19). We start with the observation that the analysis of the counterexample of Confortolaet al. [22] made in Section II.2 does not allow for a general statement of wellposedness of the BSDE when Φ≥1. In this light, the result of Cohen and Elliott [19, Theorem 6.1], which implies that the condition Φ <1 ensures the wellposedness of the BSDE, lies in the optimal range for Φ. Analogously in the case of Bandini [6], once her results are translated using the Lipschitz assumption in(F3), Φ< ^√1

2 also ensures the wellposedness of the BSDE. On the contrary, condition (II.19) which reads as Φ < 1/(18e), may seem much more restrictive. The first immediate remark we can make is that the stochasticity of the integratorC considerably deteriorates the condition on Φ. In [19] the integrator is deterministic, while in [6] and in our case the integrator is stochastic.

However, we would like to remind the reader, that, as explained above, the level of generality we are working with is substantially higher than in these two references. We also want to emphasise the fact that our condition is clearly not the sharpest one possible, but we believe it is the sharpest that can be obtained using our method of proof. The main possibilities for improvement are, in our view, twofold:

• First of all, in specific situations (e.g. T bounded,f Lipschitz, less general driving processes, . . . ) one could most probably improve thea prioriestimates of Lemma II.16 by refining several of the inequalities.

• Second, as highlighted in Remark II.17, we actually have a degree of freedom in choosing the norms we are interested in. In this paper, we used exponentials, while Bandini [6] used stochastic exponentials, but other choices, leading to potentially better estimates, could also be considered.

We leave this interesting problem of finding the optimal Φ open for future research.

CHAPTER III

Stability of Martingale Representations

This chapter deals with the stability of martingale representations. In Section I.7 we have discussed about martingale representations, therefore let us describe now roughly how do we understand their sta-bility property. For simplicity we assume that we are working on a probasta-bility space endowed with a single filtration F. Given, now, two sequences (X^k)_k∈

Nand (Y^k)_k∈

Nconsisting both of square-integrable mar-tingales, we obtain for everyk∈Nthe orthogonal decompositions ofY^kwith respect to (F, X^k,c, µ^Xk,d), say

Y^k=Z^k·X^k,c+U^k?eµ^(Xk,d,F)+N^k fork∈N. (III.1) Assuming that the sequences (X^k)_k∈

N and (Y^k)_k∈

Nare δJ₁−convergent, the martingale representations (III.1) are stable if the sequence consisting of the orthogonal martingales (N^k)_k∈Nis convergent as well as the sum of the stochastic integrals with respect to the natural pair of X^k converges to the sum of the stochastic integrals with respect to natural pair of X^∞. Schematically, given that the convergence indicated with the solid arrows hold, then also the convergence indicated with the dashed arrows hold.

X^k Y^k = Z^k·X^k,c+U^k?µe^(Xk,F) + N^k

 y

 y

99K 99K

X^∞ Y^∞ = Z^∞·X^∞,c+U^∞?µe^(X∞,F) + N^∞

Theorem III.3 demonstrates that under the suitable framework, which is presented in Section III.1, not only the martingale representations are stable, but also their associated optional as well as predictable quadratic variation processes. The former stability is a well-known result in the literature and it is a consequence of the fact that (X^k)_k∈

N, (Y^k)_k∈

N are P-UT. However, the latter stability consists a new result. We need to underline of course that it was M´emin’s Theorems, see Theorems I.159 and I.160 or [53, Theorem 11, Corollary 12], that provided us with the stability of the canonical decompositions of special semimartingales.¹

This result, although interesting on its own, is also crucial for obtaining the stability of BSDEs.

To be more precise, it will be the enhanced version of Theorem III.3, namely Theorem III.50, which we will deploy in order to obtain finally the stability of BSDEs. Commenting briefly on the connection between the martingale-representation-stability and of the BSDE-stability, the reader may recall that in the proof of Theorem II.14 we have translated the existence and uniqueness of the solution of a BSDE into a martingale representation problem. In Chapter IV we will translate the one stability problem to the othermutatis mutandis.

The structure of the chapter follows. In Section III.1 we set the framework and we present the main theorem. Then, we provide in Section III.2 the outline of the proof and in the subsequent sections we will develop the necessary machinery for the final step of the proof, which will be presented in Section III.7.

We have devoted Section III.9 for the comparison with the existing literature.

III.1. Framework and statement of the main theorem

The probability space (Ω,G,P) will be fixed throughout the chapter. The framework that we are going to set will be also valid for the whole chapter except for Section III.8. Let us, now, fix an arbitrary sequence of c`adl`agR^`−valued processes (X^k)_k∈Nfor which we assume

sup

k∈N

E Z

(0,∞)×R^`

kxk²₂µ^Xk(ds,dx)

<∞. (III.2)

1M´emin mentions in [53] that this result is not completely new, since Aldous [3] has also presented an analogous one by means of the so-calledprediction process. However, the author could not access any copy of [3] in order to provide further details on this matter.

73

Then we fix an arbitrary sequence of filtrations (G^k)_k∈

N with G^k := (G_t^k)_t≥0, for every k ∈ N, on the probability space (Ω,G,P) and an arbitrary sequence of real-valued random variables (ξ^k)_k∈N.

We make the following additional assumptions:

(M1) The filtrationG^∞is quasi-left-continuous and the processX^∞isG^∞−quasi-left-continuous.

(M2) The processX^k∈ H²(G^k;R^`) for everyk∈N.Moreover,X_∞^{k L}

2(G;R^`)

−−−−−−−→X_∞^∞. (M3) The martingale X^∞ possesses theG^∞−predictable representation property.

(M4) The filtrations converge weakly,i.e. G^k −−→^w G^∞.

(M5) The random variableξ^k ∈L²(G_∞^k ;R), for everyk∈N, andξ^{k L}

2(G∞;R)

−−−−−−−−→ξ^∞. Remark III.1. In view of Proposition I.158, conditions(M2)and(M4)imply that

(X^k,G^k)−−−→^ext (X^∞,G^∞).

Remark III.2. In(M5), we have imposed an additional measurability assumption for the sequence of random variables (ξ^k)_k∈N, since we require thatξ^k isG_∞^k −measurable for anyk∈N, instead of just being G−measurable. We could spare that additional assumption at the cost of a stronger hypothesis in(M4), namely that the weak convergence of theσ−algebrae

G_∞^k −−→ G^w _∞^∞,

holds in addition. To sum up, in the statement of Theorem III.3 the pair (M4) and (M5) can be substituted by the following

(M3⁰) The filtrations converge weakly as well as the final σ−algebrae, that is G^k −−→^w G^∞ andG_∞^k −−→ G^w _∞^∞.

(M4⁰) The sequence (ξ^k)_k∈

N⊂L²(G_∞;R) and satisfiesξ^{k L}

2(G∞;R)

−−−−−−−−→ξ^∞.

Before we state the main theorem of the chapter, it would be convenient for the reader to recall Notation I.8.

Theorem III.3. Let conditions(M1) – (M5) hold and define the G^k−martingales Y^k := Π^G_o^k(ξ^k) = E[ξ^k|G_·^k], for k∈N. Assume that the orthogonal decomposition of Y^k with respect to G^k, X^k,c, µ^Xk,d₂ is given by

Y^k =Y₀^k+Z^k·X^k,c+U^k?eµ^(Xk,d,Gk)+N^k fork∈N and

Y^∞=Y₀^∞+Z^∞·X^∞,c+U^∞?µe^{(X∞,d,G∞)} fork=∞.

Let Q^k:=Z^k·X^k,c+U^k?µe^(Xk,d,Gk) fork∈N, then the following convergence hold Y^k, Q^k, N^{k (J}1(R³),L²)

−−−−−−−−→ Y^∞, Q^∞, N^∞

, (III.3)

[Y^k],[Y^k, Q^k],[Y^k, N^k],[Y^k, X^k],[N^k] ^(J1(R³×R^`×R),L¹)

−−−−−−−−−−−−−→

[Y^∞],[Y^∞, Q^∞],[Y^∞, N^∞],[Y^∞, X^∞],[N^∞] (III.4) and

hY^ki,hY^k, X^ki,hQ^ki,hN^ki ^(J1(R×R^`×R²),L¹)

−−−−−−−−−−−−−→ hY^∞i,hY^∞, X^∞i,hQ^∞i,hN^∞i

. (III.5)

Moreover, the three above convergence can be assumed to hold jointly in D(R^9+2`). Additionally, the analogous convergence for the terminal values of the processes appear above hold.

2Observe that we regard the random measure associated to the jumps of the purely discontinuous partX^k,d of the G^k−martingaleX^k,for everyk∈N.

Im Dokument Backward stochastic differential equations with jumps are stable (Seite 73-93)